When does labor scarcity encourage innovation?

Transcription

1 When does labor scarcity encourage innovation? The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Acemogulu, Daron. "When does labor scarcity encourage innovation?" Journal of Political Economy, Vol. 118, No. 6, December University of Chicago Press Version Author's final manuscript Accessed Tue Nov 13 00:22:17 EST 2018 Citable Link Terms of Use Creative Commons Attribution-Noncommercial-Share Alike 3.0 Detailed Terms

2 When Does Labor Scarcity Encourage Innovation? Daron Acemoglu Massachusetts Institute of Technology October Abstract This paper studies whether labor scarcity encourages technological advances, i.e., technology adoption or innovation, for example, as claimed by Habakkuk in the context of 19th-century United States. I de ne technology as strongly labor saving if technological advances reduce the marginal product of labor and as strongly labor complementary if they increase it. I show that labor scarcity encourages technological advances if technology is strongly labor saving and will discourage them if technology is strongly labor complementary. I also show that technology can be strongly labor saving in plausible environments but not in many canonical macroeconomic models. JEL Classi cation: O30, O31, O33, C65. Keywords: Habakkuk hypothesis, high wages, innovation, labor scarcity, technological change. I thank Philippe Aghion, Dan Cao, Melissa Dell, Joshua Gans, Andres Irmen, Samuel Pienknagura, Su Wang, Fucheng Wei, the editor Sam Kortum, three anonymous referees, and participants at presentations at American Economic Association 2009 meetings, Harvard and Stanford for useful comments. Financial support from the National Science Foundation is gratefully acknowledged.

3 1 Introduction There is widespread consensus that technological di erences are a central determinant of productivity di erences across rms, regions, and nations. Despite this consensus, determinants of technological progress and adoption of new technologies are poorly understood. A basic question concerns the relationship between factor endowments and technology, for example, whether the scarcity of a factor, and the high factor prices that this leads to, will induce technological progress. There is currently no comprehensive answer to this question, though a large literature develops conjectures on this topic. In his pioneering work, The Theory of Wages, John Hicks was one of the rst economists to consider this possibility and argued: A change in the relative prices of the factors of production is itself a spur to invention, and to invention of a particular kind directed to economizing the use of a factor which has become relatively expensive... (1932, p. 124). Similarly, the famous Habakkuk hypothesis in economic history, proposed by H. J. Habakkuk (1962), claims that technological progress was more rapid in the 19th-century United States than in Britain because of labor scarcity in the former country, which acted as a powerful inducement for mechanization, for the adoption of labor-saving technologies, and more broadly for innovation. 1 For example, Habakkuk quotes from Pelling:... it was scarcity of labor which laid the foundation for the future continuous progress of American industry, by obliging manufacturers to take every opportunity of installing new types of labor-saving machinery. (1962, p. 6). Habakkuk continues: It seems obvious it certainly seemed so to contemporaries that the dearness and inelasticity of American, compared with British, labour gave the American entrepreneur... a greater inducement than his British counterpart to replace labour by machines. (1962, p. 17). Robert Allen (2008) has more recently argued that the relatively high wages in 18th-century Britain were the main driver of the Industrial Revolution. For example, three of the most important 18th-century technologies, Hargreaves s spinning jenny and Arkwright s water frame and carding machine, reduced labor costs in cotton manufacturing signi cantly. They were not only invented in Britain, but rapidly spread there, while their adoption was much slower in France and India. Allen (2008, Chapter 8) suggests that this was because these technologies were less pro table in France and India, where wages and thus savings in labor costs from their adoption were lower. Elvin (1972) 1 See Rothbart (1946), Salter (1966), David (1975), Stewart (1977) and Mokyr (1990) for related ideas and discussions of the Habakkuk hypothesis. 1

4 similarly suggests that a sophisticated spinning wheel used for hemp in 14th-century China was later abandoned and was not used for cotton largely because cheap and abundant Chinese labor made it unpro table. Similar ideas are often suggested as possible reasons why high wages, for example induced by minimum wages or other regulations, might have encouraged faster adoption of certain technologies, particularly those complementary to unskilled labor, in continental Europe (see, among others, Beaudry and Collard, 2002, Acemoglu, 2003, Alesina and Zeira, 2006). The so-called Porter hypothesis, which claims that tighter environmental regulations will spur faster innovation and increase productivity, is also related. 2 While this hypothesis plays a major role in various discussions of environmental policy, just like the Habakkuk hypothesis, its theoretical foundations are unclear. 3 These conjectures seem plausible at rst. Intuitions based on a downward sloping demand curve suggest that if a factor becomes more expensive, the demand for it should decrease, and we may expect some of this adjustment to take place by technology substituting for tasks previously performed by that factor. It seems compelling, for example, that technologies such as the spinning jenny, the water frame and the carding machine, which reduced the amount of labor required to produce a given quantity of cotton, should have been invented and adopted in places where the labor that they saved was more scarce and expensive. And yet, labor scarcity and high wages also reduce both the size of the workforce that may use the new technologies and the pro tability of rms, and they could discourage technology adoption through both channels. In fact, labor scarcity and high wages discourage technological advances in the most commonly-used macroeconomic models. Neoclassical growth models, when new technologies are embodied in capital goods, predict that labor scarcity and high wages slow down the adoption of new technologies. 4 Endogenous growth models also make the same prediction because lower employment discourages entry and the introduction of new technologies. 5 2 See Porter (1991) and Porter and van der Linde (1995) for the formulation of this hypothesis. Ja e, Peterson, Portney and Stavins (1995) review early empirical evidence on this topic and Newell, Ja ee and Stavins (1999) provide evidence on the e ects of energy prices on the direction of technological change. Recent work by Gans (2009) provides a theoretical explanation for the Porter hypothesis using the framework presented here. 3 Related issues also arise in the context of the study of the implications of competition from Chinese imports on technological progress. Bloom, Draca and Van Reenen (2009), for example, provide evidence that Chinese competition has encouraged innovation and productivity growth among a ected US and European rms. One of the numerous impacts of Chinese competition is to reduce employment in the a ected sectors. This creates a parallel between the aggregate impact of labor scarcity and the sectoral e ects of Chinese competition. A priori, it is not clear whether we should expect more or less investment in innovation and technology in these sectors. 4 See Ricardo (1951) for an early statement of this view. In particular, with a constant returns to scale production function F (L; K), an increase in the price of L or a reduction in its supply, will reduce equilibrium K, and to the extent that technology is embedded in capital, it will reduce technology adoption. 5 In the rst-generation models, such as Romer (1986, 1990), Segerstrom, Anant and Dinopoulos (1990), Aghion and Howitt (1992), and Grossman and Helpman (1991), it reduces the growth rate of technology and output, while 2

5 This paper investigates the impact of labor scarcity on technological advances (i.e., innovation and adoption of technologies that increase the level of output in the economy) and o ers a comprehensive answer to this question. If technology is strongly labor complementary, 6 meaning that improvements in technology increase the marginal product of labor, then labor scarcity discourages technological advances (for example, it makes such advances less likely, or in a dynamic framework, it slows down the pace of technological advances). 7 Conversely, if technology is strongly labor saving, meaning that improvements in technology reduce the marginal product of labor, then labor scarcity induces technological advances. 8 The main result in this paper can be interpreted both as a positive and a negative one. On the positive side, it characterizes a wide range of economic environments where labor scarcity can act as a force towards innovation and technology adoption, as claimed in various previous historical and economic analyses. On the negative side, it shows that this can only be so if new technology tends to reduce the marginal product of labor. This observation, in particular, implies that in most models used in the macroeconomics and growth literatures, where technological advances are assumed to increase the marginal product of labor, labor scarcity will discourage rather than induce technological advances. 9 It also implies that the relationship between labor scarcity and technological advances can vary over di erent epochs. It may well be that the technological advances of the late 18th and 19th centuries in Britain and the United States were strongly labor saving and did induce innovation and technology adoption, as envisaged by many contemporary commentators in semi-endogenous growth models, such as Jones (1995), Young (1998) and Howitt (1999), it reduces their levels. 6 The adjective strongly is added, since the terms labor complementary and labor saving are often used in several di erent contexts, not always satisfying the de nitions here. 7 More precisely, we need that aggregate output (or net output) can be expressed as a function Y (L; Z; ), where L denotes labor, Z is a vector of other factors of production, and is a vector of technologies, and Y is supermodular in, so that changes in two components of the vector do not o set each other. We say that technology is strongly labor saving if an increase in reduces the marginal product of L in Y (L; Z; ) and is strongly labor complementary if increases this marginal product. 8 Notably, in line with the directed technological change literature (e.g., Acemoglu, 1998, 2002, 2007), an increase in the supply of a factor still induces a change in technology biased towards that factor, and thus labor scarcity makes technology biased against labor. In particular, recall that a change in technology is biased towards a factor if it increases the marginal product of this factor at given factor proportions. When technology is strongly labor complementary, labor scarcity discourages technological advances and this is biased against labor. When technology is strongly labor saving, labor scarcity induces technological advances, but in this case, because there is technology-labor substitutability, this again reduces the marginal product of labor and is thus biased against labor. As a consequence, even though changes in technology in response to an increase in the supply of factor might induce or discourage technological advances, they will always biased towards that factor. 9 The fact that technological change has been the key driving force of the secular increase in wages also suggests that it may be more plausible to think of technology as strongly labor complementary rather than strongly labor saving. Nevertheless, it is possible for labor saving technology to increase wages in the long run, because past technological changes may increase wages, while current technology adoption decisions, at the margin, reduce the marginal product of labor. This is illustrated by the dynamic model presented in subsection

6 and more recently by H. J. Habakkuk and Robert Allen, 10 while this may no longer be the case in industrialized economies or even anywhere around the world. It may also be that the relevant environmental technologies have a similar substitution property with carbon, so that an increase in the price of carbon may induce more rapid advances in environmental technology (though we will also see why the reasoning is di erent in this case). I further emphasize the di erential e ects of labor scarcity by considering a multi-sector economy and showing that labor scarcity may lead to technological advances in some industries, while retarding them in others. To illustrate the implications of these results, I consider several di erent environments and production functions, and discuss when technology is strongly labor saving. An important class of models where technological change can be strongly labor saving is developed by Champernowne (1963) and Zeira (1998, 2006), and is also related to the endogenous growth model of Hellwig and Irmen (2001). In these models, technological change takes the form of machines replacing tasks previously performed by labor. I show that there is indeed a tendency of technology to be strongly labor saving in these models. Most of the analysis in this paper focuses on the implications of labor scarcity for technology choices. Nevertheless, these results can also be used to analyze the impact of an exogenous wage increase (for example, due to a minimum wage or other labor market regulation) on technology choices, because, in the context of a competitive labor market, such increases are equivalent to a decline in labor supply. 11 However, I also show the conditions under which the implications of labor scarcity and exogenous wage increases can be very di erent particularly because the long-run relationship between labor supply and wages could be upward sloping owing to general equilibrium technology e ects. Even though the investigation here is motivated by technological change and the study of economic growth, the economic environment I use for most of the paper is static. A static framework is useful because it enables us to remove functional form restrictions that would be necessary to generate endogenous growth; it thus allows the appropriate level of generality to clarify the conditions for labor scarcity to encourage innovation and technology adoption. This framework 10 This is in fact what the Luddites, who thought that new technologies would reduce demand for their labor, feared (e.g., Mokyr, 1990). Mantoux (1961) provides qualitative evidence consistent with this pattern in several industries. However, subsection 5.1 shows that even when technology is strongly labor saving, technological advances may increase wages in the long run. 11 The implications of exogenous wage increases in noncompetitive labor markets are more complex and depend on the speci c aspects of labor market imperfections and institutions. For example, Acemoglu (2003) shows that wage push resulting from a minimum wage or other labor market regulations may encourage technology adoption when there is wage bargaining and rent sharing. 4

7 is based on Acemoglu (2007) and is reviewed in Section 2. The main results of this paper and some applications are presented in Section 3. Section 4 uses several familiar models to clarify when technology is strongly labor saving. Section 5 shows how the static framework can be easily extended to a dynamic setup and also discusses other extensions, including the application to a multisector economy. Section 6 concludes. 2 The Basic Environments This section is based on and extends some of the results in Acemoglu (2007). Its inclusion is necessary for the development of the main results in Section 3. Consider a static economy consisting of a unique nal good and N + 1 factors of production. The rst factor of production is labor, denoted by L, and the rest are denoted by the vector Z= (Z 1 ; :::; Z N ) and stand for land, capital, and other human or nonhuman factors. All agents preferences are de ned over the consumption of the nal good. To start with, let us assume that all factors are supplied inelastically, with supplies denoted by L 2 R + and Z 2 R N +. Throughout I focus on comparative statics with respect to changes in the supply of labor, while holding the supply of other factors, Z, constant at some level Z (though, clearly, mathematically there is nothing special about labor). 12 The economy consists of a continuum of rms ( nal good producers) denoted by the set F, each with an identical production function. Without loss of any generality let us normalize the measure of F, jfj, to 1. The price of the nal good is also normalized to 1. I rst describe technology choices in three di erent economic environments. 13 These are: 1. Economy D (for decentralized) is a decentralized competitive economy in which technologies are chosen by rms themselves. In this economy, technology choice can be interpreted as choice of just another set of factors and the entire analysis can be conducted in terms of technology adoption. 2. Economy E (for externality) is identical to Economy D, except for a technological externality as in Romer (1986). 3. Economy M (for monopoly) will be the main environment used for much of the analysis in the remainder of the paper. In this economy, technologies are created and supplied by a pro tmaximizing monopolist. In this environment, technological progress enables the creation of better machines, which can then be sold to several rms in the nal good sector. Thus, Economy M 12 Endogenous responses of the supply of labor and other factors, such as capital, are discussed in subsections 5.4 and A fourth one, Economy O, with several technology suppliers and oligopolistic competition is discussed in the Appendix. 5

8 incorporates Romer s (1990) insight that the central aspect distinguishing technology from other factors of production is the non-rivalry of ideas. 2.1 Economy D Decentralized Equilibrium In the rst environment, Economy D, all markets are competitive and technology is decided by each rm separately. This environment is introduced as a benchmark. Each rm i 2 F has access to a production function y i = G(L i ; Z i ; i ); (1) where L i 2 R +, Z i 2 R N +, and i 2 R K is the measure of technology. 14 G is assumed to be twice continuously di erentiable and increasing in (L i ; Z i ). The cost of technology 2 in terms of nal goods is C (). This cost can be interpreted as a one-time cost that rms pay (e.g., the cost of installing new machinery), and in that case, (1) can be interpreted as representing the net present discounted value of revenues. Throughout I assume that C () is increasing in. Each nal good producer maximizes pro ts; thus, it solves the following problem: max (Li ; Z i ; i ) = G(L i ; Z i ; i ) L i ;Z i ; i w L L i NX w Zj Zj i C i, (2) j=1 where w L is the wage rate and w Zj is the price of factor Z j for j = 1; :::; N, all taken as given by the rm. The vector of prices for Z is denoted by w Z. Since there is a total supply L of labor and a total supply Z j of Z j, market clearing requires Z i2f L i di L and Z i2f Z i jdi Z j for j = 1; :::; N. (3) An equilibrium in Economy D is a set of decisions L i ; Z i ; i i2f and factor prices (w L;w Z ) such that L i ; Z i ; i i2f solve (2) given prices (w L; w Z ) and (3) holds. I refer to any i that is part of the set of equilibrium allocations, L i ; Z i ; i, as equilibrium technology. i2f In Economy D, we assume that G(L; Z; ) C () is concave in (L; Z; ). This is a restrictive assumption as it imposes concavity (strict concavity or constant returns to scale) jointly in the factors of production and technology. It is necessary for a competitive equilibrium in Economy D to exist; the other economic environments considered below will relax this assumption. 14 For most of the analysis, the reader may wish to think of as one-dimensional, though subsection 5.2 explicitly uses the multi-dimensional formulation of technology. When is multi-dimensional, we will assume that G is supermodular in and is a lattice (see, e.g., Topkis, 1998), so that di erent components of move in the same direction. 6

9 Proposition 1 Suppose that G(L; Z; ) in Economy D is a solution to C () is concave in (L; Z; ). Then equilibrium technology and any solution to this problem is an equilibrium technology. max G( L; Z; ) C () ; (4) 2 Proposition 1 implies that to analyze equilibrium technology choices, we can simply focus on a simple maximization problem. An important implication of this proposition is that the equilibrium is a Pareto optimum (and vice versa). In particular, let us introduce the notation Y denote net output in the economy with factor supplies L and Z and technology. Economy D, Y L; Z; G L; Z; C () ; and equilibrium technology maximizes net output. L; Z; to Clearly, in It is also straightforward to see that equilibrium factor prices are equal to the marginal products of the G function. That is, the wage rate is w L L; Z; )=@L and the prices of other factors are given by w Zj L; Z; )=@Z j for j = 1; :::; N, where is the equilibrium technology choice. An important implication of (4) should be emphasized. Since equilibrium technology is a maximizer of Y ( L; Z; ), any induced small change in equilibrium technology,, cannot be construed as a technological advance since it will have no e ect on net output at the starting factor proportions. In particular, assuming that Y is di erentiable in L and and that the equilibrium technology is di erentiable in L, the change in net output in response to a change in the supply of labor, L, can be written as dy ( L; Z; ) d L ( L; Z; L ( L; @ L ; (5) where the second term is the induced technology e ect. When this term is strictly negative, then a decrease in labor supply (labor scarcity) will have induced a change in technology (increasing ) that raises output. However, by the envelope theorem, this second term is equal to zero, since is a solution to (4). Therefore, there is no e ect on net output through induced technological changes, 15 and no possibility of induced technological advances because of labor scarcity in this environment (at least for small changes in technology). 16 I next consider environments with externalities or market 15 This is unless one considers changes in technology that increase output gross of costs of technology, while leaving net output unchanged, as technological advances, which does not seem entirely compelling. 16 To see the intuition for why, with competitive technology adoption, there cannot be induced technological advances, consider the comparison between British and American technologies in the 19th century discussed by Habakkuk. In the context of a fully competitive Economy D, it may have been the case that labor scarcity in the 7

10 power, where there can be induced technological advances i.e., induced changes in technology can increase net output. 2.2 Economy E Decentralized Equilibrium with Externalities The discussion at the end of the previous subsection indicated why Economy D does not enable a systematic study of the relationship between labor scarcity and technological advances (and in fact, why there is no distinction between technology and other factors of production in this economy). A rst approach to deal with this problem is to follow Romer (1986) and suppose that technology choices generate knowledge and thus create positive externalities on other rms. In particular, suppose that the output of producer i is now given by y i = G(L i ; Z i ; i ; ); (6) where is some aggregate of the technology choices of all other rms in the economy. For simplicity, we can take to be the average technology in the economy. In particular, if is a K-dimensional vector, then k = R i2f i kdi for each component of the vector (i.e., for k = 1; 2; :::; K). The remaining assumptions are the same as before. In particular, G is concave in L i ; Z i and i and increasing in L i ; Z i and. The maximization problem of each rm now becomes max (L i ; Z i ; i ; ) = G(L i ; Z i ; i ; ) L i ;Z i ; i w L L i NX w Zj Zj i C i, (7) j=1 and under the same assumptions as above, each rm will hire the same amount of all factors, so in equilibrium, L i = L and Z i = Z for all i 2 F. Then the following proposition characterizes equilibrium technology. Proposition 2 Suppose that G(L; Z; ; ) is concave in (L; Z; ). Then, equilibrium technologies in Economy E are given by the following xed point problem: 2 arg max 2 G( L; Z; ; = ) C () : (8) Even though this is a xed point problem, its structure is very similar to (4) and it can be used in the same way for our analysis (though in general multiple equilibria are possible in this United States encouraged the adoption of certain capital-intensive technologies as Habakkuk hypothesized, but the adoption of these technologies cannot be considered as technological advances since their adoption in Britain, where labor was less scarce, would have reduced rather than increased net output otherwise, they would have been adopted in Britain as well. 8

11 case). However, crucially, the envelope theorem type reasoning no longer applies to the equivalent of equation (5). To see this, let us de ne net output again as Y ( L; Z; ) G( L; Z; ; ) C (). Then once again assuming di erentiability, (5) applies, but now the second term in this expression is not equal to zero. In ( L; Z; L; Z; ; L; Z; L; Z; ; ; which is positive by assumption. This implies that induced increases in will raise output and thus correspond to induced technological advances. 2.3 Economy M Monopoly Equilibrium The main environment used for the analysis in this paper features a monopolist supplying technologies to nal good producers. There is a unique nal good and each rm has access to the production function y i = (1 ) 1 G(L i ; Z i ; ) q i () 1 ; (9) with 2 (0; 1). This expression is similar to (1), except that G(L i ; Z i ; ) is now a subcomponent of the production function, which depends on technology. The subcomponent G needs to be combined with an intermediate good embodying technology. The quantity of this intermediate used by rm i is denoted by q i () conditioned on to emphasize that it embodies technology. This intermediate good is supplied by the monopolist. The term (1 ) 1 is included as a convenient normalization. This production structure is similar to models of endogenous technology (e.g., Romer, 1990, Grossman and Helpman, 1991, Aghion and Howitt, 1992), but is somewhat more general since it does not impose that technology necessarily takes a factor-augmenting form. The monopolist can create (a single) technology 2 at cost C () from the technology menu (which is again assumed to be strictly increasing). In line with Romer s (1990) emphasis that technology has a non-rivalrous character and can thus be produced at relatively low cost once invented, I assume that once is created, the intermediate good embodying technology can be produced at constant per unit cost normalized to 1 unit of the nal good (this is also a convenient normalization). The monopolist can then set a (linear) price per unit of the intermediate good of type, denoted by. All factor markets are again competitive, and each rm takes the available technology,, and 9

12 the price of the intermediate good embodying this technology,, as given and maximizes max L i ;Z i ;q i () X N (L i ; Z i ; q i () j ; ) = (1 ) 1 G(L i ; Z i ; ) q i () 1 w L L i w Zj Zj i q i () ; which gives the following simple inverse demand for intermediates of type as a function of theirprice,, and the factor employment levels of the rm as j=1 (10) q i ; L i ; Z i j = 1 G(L i ; Z i ; ) 1= : (11) The problem of the monopolist is to maximize its pro ts: Z max = ( (1 )) q i ; L i ; Z i j di C () (12) ;;[q i (;L i ;Z i j)] i2f subject to (11). An equilibrium in Economy M is now de ned as a set of rm decisions L i ; Z i ; q i ; L i ; Z i j, technology choice and pricing decisions by the technology mo- i2f nopolist (; ), and factor prices (w L ; w Z ) such that L i ; Z i ; q i (w L ; w Z ) and (; ); (3) holds; and (; ) maximize (12) subject to (11). i2f ; L i ; Z i j i2f solve (10) given This de nition emphasizes that factor demands and technology are decided by di erent agents (the former by the nal good producers, the latter by the technology monopolist), which is an important feature both theoretically and as a representation of how technology is determined in practice. Since factor demands and technology are decided by di erent agents, we no longer require concavity of G(L i ; Z i ; ) in (L i ; Z i ; ). Instead, it is su cient that G is concave in (L i ; Z i ). 17 To characterize the equilibrium, note that (11) de nes a constant elasticity demand curve, so the pro t-maximizing price of the monopolist is given by the standard monopoly markup over marginal cost and is equal to = 1. Consequently, q i () = q i = 1; L; Z j = 1 G( L; Z; ) for all i 2 F. Substituting this into (12), the maximization problem of the monopolist can be expressed as max 2 () = G( L; Z; ) C (). Thus we have established: Proposition 3 Suppose that G(L; Z; ) is concave in (L; Z) (for all 2 ). Then any equilibrium technology in Economy M is a solution to and any solution to this problem is an equilibrium technology. max G( L; Z; ) C () ; (13) 2 17 There is no loss of generality if G, is taken to exhibit constant returns to scale in L and Z in the rest of the analysis. 10

13 This proposition shows that equilibrium technology in Economy M is a solution to a problem identical to that in Economy D, that of maximizing G( L; Z; ) C () as in (4). Naturally, the presence of the monopoly markup introduces distortions in the equilibrium. These distortions are the reason why equilibrium technology is not at the level that maximizes net output. In particular, let us use the fact that the pro t-maximizing monopoly price is = 1 and substitute (11) into the production function (9), and then subtract the cost of technology choice, C (), and the cost of production of the machines, (1 this economy as ) 1 G(L i ; Z i ; ), from gross output. This gives net output in Y L; Z; 2 1 G L; Z; C () : (14) Clearly, the coe cient in front of G L; Z; is strictly greater than 1. Recall also that C is strictly increasing in, and thus in any interior equilibrium, G must also be strictly increasing in. This implies that, as in Economy E, Y equilibrium. L; Z; will be increasing in in the neighborhood of any Finally, it can be veri ed that equilibrium factor prices are given by w L = (1 ) L; Z; )=@L and w Zj = (1 ) L; Z; )=@Z j and are also proportional to the derivatives of the net output function Y de ned in (14). In what follows, I take Economy M as the baseline Labor Scarcity and Technological Progress This section presents the main results of the paper and a number of extensions and applications. 18 Using this framework, Acemoglu (2007) investigates the question of (induced) equilibrium bias of technology i.e., whether an increase in the supply of a factor, say labor L, will change technology in a way that is weakly or strongly equilibrium biased towards L. We say that there is weak equilibrium bias if the combined e ect of induced changes in technology resulting from an increase in labor supply raise the marginal product of labor at the starting factor proportions (i.e., it shifts out the demand for labor). Similarly, there is strong equilibrium bias if this induced e ect in technology is su ciently large so as to outweigh the direct e ect of the increase in L (which is always to reduce its marginal product). The results in that paper show that there is always weak equilibrium bias, meaning that any increase in the supply of a factor always induces a change in technology favoring that factor. Moreover, this e ect can be strong enough so that there is strong equilibrium bias, in which case, in contrast to basic producer theory, endogenous technology choices in general equilibrium will lead to upward sloping demand curves for factors. More speci cally, there will be strong equilibrium bias if and only if the Hessian of the production function with respect to L and, r 2 F (L;)(L;), is not negative semi-de nite (see Theorem 6 in the Appendix). Since in economies M and D, L and are chosen by di erent agents, there is no presumption in general that r 2 F (L;)(L;) need to be negative semi-de nite. Interestingly, these results about equilibrium bias imply almost nothing about the impact of labor scarcity on technological advances as a change in technology biased towards a factor could correspond to either a technological advance or a deterioration in technology. 11

14 3.1 Main Result Let us focus on Economy M in this subsection and impose the following assumption to simplify the exposition. Assumption 1 Let = R K +. C () is twice continuously di erentiable, strictly increasing and strictly convex in 2, and for each k = 1; 2; :::; K, we () lim = 0 and lim = 1 for all. k k Moreover, G L; Z; is continuously di erentiable in and L, and concave in 2, and satis L; lim Z; > 0 for all and Z. k Recall that equilibrium technology, L; Z, is a solution to the maximization problem in (13). Assumption 1 then ensures equilibrium technology i.e., it satis es L; Z is uniquely determined and L; Z; L; Z L; Z for k = 1; 2; :::; k Moreover, in this equilibrium, it must be the case L; Z; L; Z =@k > 0 (for each k = 1; 2; : : : ; K) as C L; Z is strictly increasing from Assumption 1. Since net output Y L; Z; is given by (14), this also implies L; Z; L; k > 0 for k = 1; 2; :::; K: (15) In light of this, we say that there are technological advances if increases (meaning that each component of the vector increases or remains constant). The key concepts of strongly labor (or more generally factor) saving technology and strongly labor complementary technology are introduced in the next de nition. Let x = (x 1 ; :::; x n ) in R n. Then recall that a twice continuously di erentiable function f (x) is supermodular on X if and only 2 f (x) =@x i 0 0 for all x 2 X and for all i 6= i 0. In addition, a function f (x; t) de ned on X T (where X R n and T R m ) has increasing di erences in (x; t) if for all t 00 > t, f (x; t 00 ) f (x; t) is nondecreasing in x and has strict increasing di erences in (x; t), if for all t 00 > t, f (x; t 00 ) f (x; t) is increasing in x. 19 Decreasing di erences and strict decreasing di erences are de ned analogously by f (x; t 00 ) f (x; t) being nonincreasing and decreasing, respectively. If f is di erentiable and T R, then increasing di erences is equivalent 2 f (x; t) =@x 0 for each i and decreasing di erences is equivalent 2 f (x; t) =@x 0 for each i. 19 Throughout, increasing stands for strictly increasing, and decreasing for strictly decreasing. 12

15 De nition 1 Technology is strongly labor saving at L, Z; if there exist neighborhoods BL, B Z and B of L, Z and such that G (L; Z; ) exhibits strict decreasing di erences in (L; ) on B L B Z B. Conversely, technology is strongly labor complementary at L, Z; if there exist neighborhoods B L, B Z and B of L, Z and such that G (L; Z; ) exhibits strict increasing di erences in (L; ) on B L B Z B. We say that technology is strongly labor saving [resp., complementary] globally if it is strongly labor saving [resp., complementary] for all L, Z and 2. Intuitively, technology is strongly labor saving if technological advances reduce the marginal product of labor, and it is strongly labor complementary if technological advances increase the marginal product of labor. The next theorem gives a fairly complete characterization of when labor scarcity will induce technological advances. Theorem 1 Consider Economy M and suppose that Assumption 1 holds and G (L; Z; ) C () is supermodular in. Let the equilibrium technology be denoted by L, Z. Then labor scarcity will induce technological advances (increase ), in the sense k L, Z =@ L < 0 for each k = 1,...,K, if technology is strongly labor saving at L, Z; L, Z, and will discourage technological advances, in the sense k L, Z =@ L > 0 for each k = 1,...,K, if technology is strongly labor complementary at L, Z; L, Z. Proof. From Assumption 1, G is increasing in in the neighborhood of L, Z. Equation (15) then implies that technological advances correspond to a change in technology from 0 to From Assumption 1, (13) is strictly concave and the solution L, Z is strictly positive, unique and, by the implicit function theorem, di erentiable in L. Therefore, a small change in L will lead to a small change in each of k L, Z (k = 1,...,K). Since G (L; Z; ) C () is supermodular in by assumption, comparative statics are determined by whether G exhibits strict decreasing or increasing di erences in L and in the neighborhood of L, Z, and L, Z. In particular, Theorem in Topkis (1998) implies that when technology is strongly labor saving, i.e., when G exhibits strict decreasing di erences in L k L, Z =@ L < 0 for each k = 1,...,K. This yields the result for strongly labor saving technology. Conversely, when G exhibits strict increasing di erences in L k L, Z =@ L > 0 for each k = 1,...,K, and labor scarcity reduces, establishing the desired result for strongly labor complementary technology. Though simple, this theorem provides a fairly complete characterization of the conditions under which labor scarcity will lead to technological advances. The only cases that are not covered by 13

16 the theorem are those where G is not supermodular in and those where G exhibits neither increasing di erences nor decreasing di erences in L and. Without supermodularity, the direct e ect of labor scarcity on each technology component would be positive, but because of lack of supermodularity, the advance in one component may then induce an even larger deterioration in some other component, thus a precise result becomes impossible. When G exhibits neither increasing or decreasing di erences, then a change in labor supply L will a ect di erent components of technology in di erent directions, and without making further parametric assumptions, we cannot reach an unambiguous conclusion about the overall e ect. Clearly, when is single dimensional, the supermodularity condition is automatically satis ed, and G exhibits either increasing or decreasing di erences in the neighborhood of L, Z and L, Z (recall that when is single dimensional, decreasing di erences is equivalent 2 G=@L@ 0 and increasing di erences 2 G=@L@ 0). Another potential shortcoming of this analysis is that the environment is static. Although these results are stated for a static model, there are multiple ways of extending this framework to a dynamic environment and the main forces will continue to apply in this case (see subsection 5.1 for an illustration of this point using an extension to a growth model). The advantage of the static environment is that it enables us to develop these results at a fairly high level of generality, without being forced to make functional form assumptions in order to ensure balanced growth or some other notion of a well-de ned dynamic equilibrium. 3.2 Further Results The results of Theorem 1, which were stated under Assumption 1 and for Economy M, can be generalized to Economy E and they can also be extended to global results. The next theorem provides the analog of Theorem 1 for Economy E, except that now equilibrium technology need not be unique (since the equilibrium is a solution to a xed point problem rather than a maximization problem). As is well known (e.g., Milgrom and Roberts, 1994, Topkis, 1998), when there are multiple equilibria, we can typically only provide unambiguous comparative statics for extremal equilibria. These extremal equilibria always exist in the present context given the assumptions we have imposed so far (supermodularity of G and the fact that is a lattice), and they correspond to the smallest and greatest equilibrium technologies, and + (meaning that if there exists another equilibrium technology, ~, we must have + ~ ). In view of this, a technological advance now refers to an increase in the greatest and the smallest equilibrium technologies. Theorem 2 Consider Economy E, and suppose that Assumption 1 holds and also that 14

17 G L; Z; ; C () is supermodular in and. Let and + denote the smallest and the greatest equilibrium technologies at L, Z. Then if technology is strongly labor saving at L, Z; [resp., at L, Z; + ] labor scarcity will induce technological advances (in the sense that a small decrease in L will increase [resp., + ]), if technology is strongly labor complementary at L, Z; [resp., at L, Z; + ] labor scarcity will discourage technological advances (in the sense that a small decrease in L will reduce [resp., + ]). Proof. See the Appendix. We next present global versions of Theorems 1 and 2, which hold without Assumption 1 when technology is strongly labor saving or labor complementary globally. The statements again refer to the smallest and the greatest equilibria. Theorem 3 Consider Economy M or E. Suppose that Assumption 1 holds and G (L; Z; ) C () is supermodular in in Economy M or G L; Z; ; C () is supermodular and increasing in in Economy E. If technology is strongly labor saving [labor complementary] globally, then labor scarcity will induce [discourage] technological advances in the sense of increasing [reducing] the smallest and the greatest equilibrium technologies, and +. Proof. I provide the proof for Economy E (the proof for Economy M is similar but more straightforward as the equilibrium is still a solution to a maximization problem). When G exhibits increasing di erences in L and globally, the payo of each rm i exhibits increasing di erences in its own strategies and L. Then, given that G is supermodular in 0 and, Theorem from Topkis (1998) implies that the greatest and smallest equilibria of this game are nondecreasing in L and Assumption 1 again guarantees that equilibria are interior and thus must be increasing in L. This establishes the second part of the theorem. The rst part follows with the same argument, using instead of, when technology is strongly labor saving globally. This theorem shows that similar results hold for Economy M or E (and the Appendix shows that they also extend to an oligopolistic setting). It can also be shown that similar changes in also hold in Economy D. But for reasons already emphasized, increases in in Economy D do not correspond to technological advances because in the neighborhood of an equilibrium in Economy D, any change will reduce net output at given L and Z, and small changes will have second-order e ects in the neighborhood of L and Z because equilibrium technology maximizes output at these factor proportions Although the statement may not be true for non-in nitesimal changes, it is immediate consequence of Proposition 1 that any (induced) change in starting from cannot increase net output at L and Z. The only reason why 15

18 3.3 Implications of Exogenous Wage Increases Let us de ne F (L; Z; ) G(L; Z; ) C () and let r 2 F (L;)(L;) denote the Hessian of this function with respect to L and. The Appendix (in particular Theorem 6) shows that if r 2 F (L;)(L;) is negative semi-de nite, the relationship between employment and the equilibrium wage, even in the presence of endogenous technology, is given by a decreasing function wl (L). As a consequence, we can equivalently talk of a decrease in labor supply (corresponding to labor becoming more scarce ) or an exogenous wage increase, where a wage above the market clearing level is imposed. In this n light, we can generally think of equilibrium employment as L e = min (wl ) 1 (wl e ) ; L o, where w e L is the equilibrium wage rate, either determined in competitive labor markets or imposed by regulation. Under these assumptions, all of the results presented in this section continue to hold. This result is stated in the next corollary. Corollary 1 Suppose that r 2 F (L;)(L;) is negative semi-de nite. Then under the same assumptions as in Theorems 1-3, a minimum wage above the market clearing wage level induces technological advances when technology is strongly labor saving and discourages technological advances when technology is strongly labor complementary. Proof. Theorem 6 in the Apendix implies that when r 2 F (L;)(L;) is negative semi-de nite, a wage above the market clearing level is equivalent to a decline in employment. Then the result in the corollary follows from Theorems 1 and 3. The close association between labor scarcity and exogenous wage increases in this result relies on the assumption that r 2 F (L;)(L;) is negative semi-de nite, so that the endogenous-technology demand curves are downward sloping (recall Theorem 6). When this is not the case, exogenous wage increases can have richer e ects and this is discussed in subsection 5.4. While Corollary 1 shows that exogenous wage increases can induce technological advances, it should be noted that even when this is the case, net output may decline because of the reduction in employment. 21 Nevertheless, when the e ect of labor scarcity on technology is su ciently pronounced, overall output may increase even though employment declines. Consider the following example, which illustrates both this possibility and also gives a simple instance where technology is strongly labor saving. caution is necessary is that such a change, while reducing net output at L and Z, may increase it at some other factor proportions. 21 Conversely, even if labor scarcity does not encourage technological advances, output per worker might increase because of the standard channel of diminishing returns to labor. 16

19 Example 1 Let us focus on Economy M, and suppose that Z = (K; T ) (where K denotes capital and T land), and the G function takes the form G (L; K; T; ) = 3 K 1=3 + (1 ) L 1=3 T 2=3 ; and the cost of technology creation is C () = 3 2 =2. Intuitively, here is a technology that shifts tasks away from labor towards capital (see subsection 4.3). Let us normalize the supply of the non-labor factors to K = T = 1 and denote labor supply by L. Suppose equilibrium wages are given by the marginal product of labor. The equilibrium technology is L = 1 L 1=3. The equilibrium wage, the marginal product of labor at L and technology, is then w L; = (1 ) L 2=3 : To obtain the endogenous technology relationship between labor supply and wages, we substitute L into this wage expression and obtain w L; (L) = L 1=3 ; which shows that there is a decreasing relationship between labor supply and wages. Suppose that labor supply L is equal to 1=64. In that case, the equilibrium wage will be 4. Next consider a minimum wage at w = 5. Since nal good producers take prices as given, they have to be along their (endogenous-technology) labor demands; this implies that employment will fall to L e = 1=125. Without the exogenous wage increase, technology was L = 3=4, whereas after the minimum wage, we have (L e ) = 4=5, which illustrates the induced technology adoption/innovation e ects of exogenous wage increases. Do such wage increases increase overall output? Recall that net output is equal to Y (L; Z; ) (2 ) = (1 ) G (L; Z; ) C (), where 1 is the share of intermediates in the nal good production function (recall equation (9)). It can be veri ed that for close to 0, an exogenous wage increase reduces net output; however for su ciently close to 1, net output increases despite the decline in employment. Generalizing this example, it can be veri ed that when R, an exogenous wage increase will increase output if the following conditions are satis ed: (1) technology is strongly labor saving; L; Z; 2 G L; Z; =@ 2 2 G L; Z; L; Z; =@; and (3) is su ciently close to 1. These conditions can be easily generalized to cases in which is multidimensional. 17

20 3.4 Applications In this subsection, we brie y discuss two applications: the implications of carbon taxes for green technology, and the impacts of scarcity of skilled and unskilled labor. 22 It is straightforward to apply the framework developed so far to investigate the Porter hypothesis discussed in the Introduction. 23 To do this, let us focus on Economy M, with the only di erence being that corresponds to green technologies and p, which represents carbon or pollution, replaces L (for simplicity, we are ignoring non-green technologies). Note, however, that p is not an input, but part of the joint output. Thus output is given by (9) with G Z i ; replacing G L i ; Z i ;, and pollution is given as p = (1 ) 1 P (Z i ; ) q i () 1 ; where the function P (Z; ) is assumed to be decreasing in, capturing the fact that is a vector of green technologies, and (1 ) 1 is again included as a normalization. We then assume that the cost of introducing technology, C (), is increasing, capturing the fact that more green technologies are more expensive. Final good producers pay a tax equal to units of nal good on their production of p. It is then straightforward to see that, instead of (11), the demand for machines from the nal good sector will be given by q i ; Z i j = 1 G(Z i ; ) P (Z i ; ) 1= 1= ; where again denotes the per unit price of machines embedding technology. This expression simply follows from the fact that the net revenue of the rm is now proportional to G(Z i ; ) P (Z i ; ) 1=. An equilibrium is de ned in a similar fashion, except that will be a solution to max G(Z i ; ) P (Z i ; ) 1= 2 C () : Consider now an increase in environmental regulation, captured by a higher tax on pollution or carbon, i.e., higher. Since P is decreasing in, this will clearly increase the marginal return to, and will increase. But this does not imply that environmental regulation will encourage 22 Gans (2009) also uses the framework developed in this paper to investigate the Porter hypothesis and Acemoglu et al. (2010) develop a two-sector economy with directed technical change and dynamic environmental externalities to study the implications of environmental regulations on technological change and climate. 23 It should be noted that what is being discussed here is a sophisticated Porter hypothesis. Porter s (1995) article implies that regulation on a single rm can increase that rm s pro tability, which is not possible provided that rms are maximizing (net present discounted value of) pro ts. However, regulation or taxes on an industry can increase each rm s pro tability, which is the sophisticated version of the hypothesis discussed here (without adding this quali er in what follows to simplify the terminology). 18